In 1892, Lord Rayleigh estimated the effective conductivity of rectangular arrays of disks and proved, employing the Eisenstein summation, that the lattice sum \(S_2\) is equal to \(\pi \) for the square array. Further, it became clear that such equality can be treated as a necessary condition of the macroscopic isotropy of composites governed by the Laplace equation. This yielded the description of two-dimensional conducting composites by the classic elliptic functions, including the conditionally convergent Eisenstein series. In 1935, Natanzon used a polyharmonic function to solve the plane elasticity problem. This paper is devoted to the extension of the classic lattice sums to the lattice sums for double periodic (pseudoperiodic) polyanalytic functions. The exact relations and computationally effective formulae between the polyanalytic and classic lattice sums are established. Polynomial representations of the lattice sums are obtained. They are a source of new exact formulae for the lattice sums.

1892年，瑞利勋爵估计了圆盘矩形阵列的有效电导率，并利用爱森斯坦求和证明了晶格和\(S_2\)等于\(\pi \)对于方阵。此外，很明显，这种等式可以被视为拉普拉斯方程控制的复合材料宏观各向同性的必要条件。这产生了通过经典椭圆函数（包括条件收敛的爱森斯坦级数）对二维导电复合材料的描述。1935年，Natanzon使用多调和函数解决了平面弹性问题。本文致力于将经典格和扩展到双周期（伪周期）多解析函数的格和。建立了多解析和经典格和之间的精确关系和计算有效的公式。获得晶格和的多项式表示。它们是新的精确格和公式的来源。